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Theorem dvdsval2 13849
Description: One nonzero integer divides another integer if and only if their quotient is an integer. (Contributed by Jeff Hankins, 29-Sep-2013.)
Assertion
Ref Expression
dvdsval2  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( N  /  M )  e.  ZZ ) )

Proof of Theorem dvdsval2
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 divides 13848 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  E. k  e.  ZZ  (
k  x.  M )  =  N ) )
213adant2 1015 . 2  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  E. k  e.  ZZ  ( k  x.  M )  =  N ) )
3 zcn 10868 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  CC )
433ad2ant3 1019 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  N  e.  CC )
54adantr 465 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  N  e.  CC )
6 zcn 10868 . . . . . . . . . 10  |-  ( k  e.  ZZ  ->  k  e.  CC )
76adantl 466 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  k  e.  CC )
8 zcn 10868 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  CC )
983ad2ant1 1017 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  M  e.  CC )
109adantr 465 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  M  e.  CC )
11 simpl2 1000 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  M  =/=  0
)
125, 7, 10, 11divmul3d 10353 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  ( ( N  /  M )  =  k  <->  N  =  (
k  x.  M ) ) )
13 eqcom 2476 . . . . . . . 8  |-  ( N  =  ( k  x.  M )  <->  ( k  x.  M )  =  N )
1412, 13syl6bb 261 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  ( ( N  /  M )  =  k  <->  ( k  x.  M )  =  N ) )
1514biimprd 223 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  ( ( k  x.  M )  =  N  ->  ( N  /  M )  =  k ) )
1615impr 619 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( k  e.  ZZ  /\  ( k  x.  M )  =  N ) )  -> 
( N  /  M
)  =  k )
17 simprl 755 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( k  e.  ZZ  /\  ( k  x.  M )  =  N ) )  -> 
k  e.  ZZ )
1816, 17eqeltrd 2555 . . . 4  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( k  e.  ZZ  /\  ( k  x.  M )  =  N ) )  -> 
( N  /  M
)  e.  ZZ )
1918rexlimdvaa 2956 . . 3  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( E. k  e.  ZZ  ( k  x.  M
)  =  N  -> 
( N  /  M
)  e.  ZZ ) )
20 simpr 461 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( N  /  M )  e.  ZZ )  ->  ( N  /  M )  e.  ZZ )
21 simp2 997 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  M  =/=  0 )
224, 9, 21divcan1d 10320 . . . . . 6  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  (
( N  /  M
)  x.  M )  =  N )
2322adantr 465 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( N  /  M )  e.  ZZ )  ->  ( ( N  /  M )  x.  M )  =  N )
24 oveq1 6290 . . . . . . 7  |-  ( k  =  ( N  /  M )  ->  (
k  x.  M )  =  ( ( N  /  M )  x.  M ) )
2524eqeq1d 2469 . . . . . 6  |-  ( k  =  ( N  /  M )  ->  (
( k  x.  M
)  =  N  <->  ( ( N  /  M )  x.  M )  =  N ) )
2625rspcev 3214 . . . . 5  |-  ( ( ( N  /  M
)  e.  ZZ  /\  ( ( N  /  M )  x.  M
)  =  N )  ->  E. k  e.  ZZ  ( k  x.  M
)  =  N )
2720, 23, 26syl2anc 661 . . . 4  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( N  /  M )  e.  ZZ )  ->  E. k  e.  ZZ  ( k  x.  M
)  =  N )
2827ex 434 . . 3  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  (
( N  /  M
)  e.  ZZ  ->  E. k  e.  ZZ  (
k  x.  M )  =  N ) )
2919, 28impbid 191 . 2  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( E. k  e.  ZZ  ( k  x.  M
)  =  N  <->  ( N  /  M )  e.  ZZ ) )
302, 29bitrd 253 1  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( N  /  M )  e.  ZZ ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   class class class wbr 4447  (class class class)co 6283   CCcc 9489   0cc0 9491    x. cmul 9496    / cdiv 10205   ZZcz 10863    || cdivides 13846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-z 10864  df-dvds 13847
This theorem is referenced by:  dvdsval3  13850  nndivdvds  13852  fsumdvds  13887  3dvds  13908  bitsmod  13944  sadaddlem  13974  bitsuz  13982  mulgcd  14042  sqgcd  14054  prmind2  14086  mulgcddvds  14103  qredeu  14106  isprm5  14111  divgcdodd  14118  divnumden  14139  hashdvds  14163  oddprm  14197  pythagtriplem11  14207  pythagtriplem13  14209  pythagtriplem19  14215  pcprendvds2  14223  pcpremul  14225  pc2dvds  14260  pcz  14262  pcadd  14266  pcmptdvds  14271  fldivp1  14274  pockthlem  14281  prmreclem1  14292  prmreclem3  14294  4sqlem8  14321  4sqlem9  14322  4sqlem12  14332  4sqlem14  14334  sylow1lem1  16421  sylow3lem4  16453  odadd1  16654  odadd2  16655  pgpfac1lem3  16927  prmirredlem  18306  prmirredlemOLD  18309  znidomb  18383  root1eq1  22873  atantayl2  23013  efchtdvds  23177  dvdsdivcl  23201  muinv  23213  chtub  23231  bposlem6  23308  lgseisenlem1  23368  lgsquad2lem1  23377  lgsquad3  23380  m1lgs  23381  2sqlem3  23385  2sqlem8  23391  qqhval2lem  27614  nn0prpwlem  29733  congrep  30531  jm2.22  30557  jm2.23  30558  hashgcdlem  30778  proot1ex  30782  lcmgcdlem  30828  nzss  30838
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